More generally, when considering polynomials, it is observed that \(S^{k}x(t)\)  is equivalent to exchanging \(k\)  times information with one-hop neighbors.
Therefore, if the system matrix \(A\)  and the control matrix \(B\)  are polynomials of \(S\) , then the evolution of the system can be computed entirely by means of exchanges with neighboring nodes. Hence, the system dynamics can be viewed as implementing distributed message passing [1]}.
Examples of such linear, distributed systems, include both discrete-time and continuous-time diffusions, solutions to the heat equation, among many others, see [2]} and references therein.
